## Name for "Boolean algebra (logic)"

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• What should a Wikipedia article on the syntactic (equational deduction) aspects of Boolean algebra be called? Currently there are a number of articles related
Message 1 of 7 , Jul 10, 2007
What should a Wikipedia article on the syntactic (equational deduction)
aspects of Boolean algebra be called?

Currently there are a number of articles related in one way or another
to Boolean algebra. It is currently proposed on the Boolean algebra
discussion page to have two articles, one on Boolean algebras (the
objects of the variety) which already exists under the title "Boolean
algebra" (not as fast-paced or comprehensive as Donald Monk's very nice
article at http://plato.stanford.edu/entries/boolalg-math/), and one on
more syntactic aspects (relationships between operation bases,
equational reasoning, etc.). Such a split is reasonable to permit more
material overall without the individual articles getting too unwieldy.

Possible names for the syntactic article include

1. Boolean logic

2. Boolean algebra (logic)

3. Elementary Boolean algebra

4. Symbolic Boolean algebra

1 and 3 already exist as Wikipedia titles, at respectively
http://en.wikipedia.org/wiki/Boolean_logic and
http://en.wikipedia.org/wiki/Elementary_Boolean_algebra . No one seems
particularly happy with 1. I wrote 3 with an eye to eventually
substituting it for at least some of 1. The name 2 has been proposed by
some, to go along with a renaming of "Boolean algebra" to "Boolean
algebra (structure)" by way of disambiguation.

What would be your preferred name for the syntactic article (not
necessarily one of the above four)? A matching name for the algebras
article would also be nice, but the main question for now is with the other.

Vaughan Pratt
• I forgot to say that a common though not consistently used Wikipedia convention for disambiguating two topics naturally named the same is a parenthetical
Message 2 of 7 , Jul 10, 2007
I forgot to say that a common though not consistently used Wikipedia
convention for disambiguating two topics naturally named the same is a
parenthetical disambiguator, as in

Ring (mathematics)
Ring (computer security)

So one possible matched pair in that format could be

Boolean algebra (elementary)
Boolean algebra (abstract)

Or perhaps "Boolean algebras (variety)" for the second.

Also, while new names (if good) are helpful, what won't be helpful is to
end up with a list of names each with exactly one expression of support.

Vaughan
• I vote for Boolean algebra (logic) . B ____________________________________________ Dr Brian A. Davey Reader and Associate Professor Department of
Message 3 of 7 , Jul 10, 2007
I vote for "Boolean algebra (logic)".

\B

____________________________________________

Dr Brian A. Davey
Department of Mathematics
La Trobe University
Victoria 3086
Australia
Phone: +61 3 9479 2599 (Office)   +61 3 9479 2600 (Sec.)
FAX: +61 3 9479 2466
Email: B.Davey@...
http://www.latrobe.edu.au/www/mathstats/staff/davey/
____________________________________________

From: univalg@yahoogroups.com [mailto:univalg@yahoogroups.com] On Behalf Of Vaughan Pratt
Sent: Tuesday, 10 July 2007 6:42 PM
To: univalg@yahoogroups.com
Subject: [univalg] Name for "Boolean algebra (logic)"

What should a Wikipedia article on the syntactic (equational deduction)
aspects of Boolean algebra be called?

Currently there are a number of articles related in one way or another
to Boolean algebra. It is currently proposed on the Boolean algebra
discussion page to have two articles, one on Boolean algebras (the
objects of the variety) which already exists under the title "Boolean
algebra" (not as fast-paced or comprehensive as Donald Monk's very nice
article at http://plato. stanford. edu/entries/ boolalg-math/), and one on
more syntactic aspects (relationships between operation bases,
equational reasoning, etc.). Such a split is reasonable to permit more
material overall without the individual articles getting too unwieldy.

Possible names for the syntactic article include

1. Boolean logic

2. Boolean algebra (logic)

3. Elementary Boolean algebra

4. Symbolic Boolean algebra

1 and 3 already exist as Wikipedia titles, at respectively
http://en.wikipedia .org/wiki/ Boolean_logic and
http://en.wikipedia .org/wiki/ Elementary_ Boolean_algebra . No one seems
particularly happy with 1. I wrote 3 with an eye to eventually
substituting it for at least some of 1. The name 2 has been proposed by
some, to go along with a renaming of "Boolean algebra" to "Boolean
algebra (structure)" by way of disambiguation.

What would be your preferred name for the syntactic article (not
necessarily one of the above four)? A matching name for the algebras
article would also be nice, but the main question for now is with the other.

Vaughan Pratt

• ... I have the following opinion based on the citations above. (A) I think, 1. Boolean logic would be one of the right titles, since it is an important
Message 4 of 7 , Jul 11, 2007
> 1. Boolean logic
> 2. Boolean algebra (logic)

> 1 and 3 already exist as Wikipedia titles, at respectively
> http://en.wikipedia.org/wiki/Boolean_logic and
> http://en.wikipedia.org/wiki/Elementary_Boolean_algebra .

> No one seems particularly happy with 1. I wrote 3 with an eye to
> eventually substituting it for at least some of 1.

> The name 2 has been proposed by some, to go along with a renaming of
> "Boolean algebra" to "Boolean algebra (structure)" by way of
> disambiguation.

I have the following opinion based on the citations above.

(A) I think, "1. Boolean logic" would be one of the right titles, since it
is an important attribute of a particular logic, whether a statement can
have one complement only or more (non-distributive logic).

(B) Pragmatically accepting that one the right titles in question is
already occupied, it is a bright suggestion to use

"2. Boolean algebra (logic)" and

"?. Boolean algebra (structure)"

in parallel for the two subtopics in question.

Perphaps, in the case of "2. Boolean algebra (logic)" the much more
precise "Boolean logic" should be used as a subtitle or topic
classification, in order to help the diverse search engines to find the
right page.

Peter.
• Roughly speaking... (I mean, disregarding a lot of subtle points) Bourbaky classified the mathematical structures in three categories: algebraic, order, and
Message 5 of 7 , Jul 13, 2007
Roughly speaking... (I mean, disregarding a lot of subtle points)

Bourbaky classified the mathematical structures in three categories: algebraic, order, and topological.
Beziau introduced a fourth category: logical.

Boolean algebras can be seen, without pain, as living inside the four universes above.
Moreover, there are easy ways to transform a BA viewed as an algebraic structure into a BA viewed as an oder structure into a BA viewed as a topological structure into BA viewed as a logical structure, and back.
To me, seems that algebraic, order and topological BAs are closer to each other than to logical BAs.
So I vote with Peter:

1. Boolean algebra (structure)

2. Boolean algebra (logic)

best regards,
--
Petrucio Viana

Departamento de Análise
Instituto de Matemática

e-mail: petrucio@...
web page: http://www.cos.ufrj.br/~petrucio

• ... Initially I was sceptical that these constituted distinct notions---my reaction was, aren t algebraic, order, and logical BAs all the same? (Topological
Message 6 of 7 , Jul 15, 2007
> Posted by: "Jorge Petrúcio Viana" petrucio@...
> Fri Jul 13, 2007 4:23 am (PST)
>
> To me, seems that algebraic, order and topological BAs are closer to
> each other than to logical BAs.

Initially I was sceptical that these constituted distinct notions---my
reaction was, aren't algebraic, order, and logical BAs all the same?
(Topological BAs are different of course, being those BAs with a
topology and continuous homomorphisms.)

However if one views an order BA as a poset with the requisite sups and
infs satisfying the requisite laws, taking the signature to be that of a
poset with no lattice operations, one could reasonably argue that the
morphisms should be just all monotone functions. Is there any other
notion of order BA that gives it a different extension from the
algebraic or ordinary BAs?

In the same vein, one could reasonably take the logical BAs to be either
the initial BA (the prototypical BA defining the equational theory) or
the free BAs (although one might question the value to logic of the
uncountably generated free BAs).

The method in the apparent madness of the different rules for
topological and order BAs (morphisms respecting the meets and joins of
the former but not the latter) can be illustrated with the example of
CABAs, the complete atomic BAs. Since CABAs are automatically
topological BAs, the above reasoning distinguishing order BAs from
ordinary BAs when applied to CABAs should result in topological CABAs
being distinguished from ordinary CABAs by taking their signature to
exclude the sups and infs and to contain only the topology and the order
(the latter being subsumable by the topology only if one gives up the
convention that a CABA is a totally order disconnected space) and hence
the morphisms to be the continuous monotone functions. Topological
CABAs follow a different rule than topological BAs for the same reason
order BAs follow a different rule than ordinary BAs: if they followed
the same rule topological CABAs would merely be CABAs, just as order BAs
would merely be BAs, and hence a waste of a potentially useful term.

None of this however seems to support Jorge's sense that the algebraic,
order, and topological BAs form a natural cluster separate from the
logical BAs. On the other hand one can at least make the argument that,
among all adjunctions F |- U for which UF is the monad (equational
theory) of Boolean algebras, the logical and ordinary BAs are at the
distal end from Set of the initial and final such adjunctions, being
respectively the Kleisli and Eilenberg-Moore categories. In that sense
they are as far apart as they can get. Absent an equally compelling
force separating order and topological BAs from ordinary BAs, perhaps
the natural gravity of mathematics pulls these three bodies into a tight
but chaotic orbit around each other.

We now know by hard experience to check the radars of journals. Lattice
theory being completely off that of the Journal of Irreproducible
Results, it would be irresponsible to waste its referees' time by
submission of any substantive development of the above theme. As Arthur
C. Clarke would have put it, any sufficiently advanced mathematics is
indistinguishable from gibberish.

Vaughan Pratt
• Dear all, I was recently given the following link, which I read with much enjoyment: http://www.math.rutgers.edu/~zeilberg/Opinion81.html Many congratulations
Message 7 of 7 , Aug 26, 2007
Dear all,

I was recently given the following link, which I read with much enjoyment:

Many congratulations again to our 4-member Dilworth committee!

Cheers, Fred

--
Friedrich Wehrung
LMNO, CNRS UMR 6139
Universit\'e de Caen, Campus 2
D\'epartement de Math\'ematiques, BP 5186
14032 Caen cedex
FRANCE

e-mail: wehrung@...
alternate e-mail: fwehrung@...

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